Some Properties of Operations on $\Alpha O(X)$

Topological Group Via Operations defined on α-Open Sets

“…The αCl β -closure of A is denoted by αCl β (A). The union of all α β -open sets contained in A is called the α β -interior of A and denoted by α β Int(A) [2]. An operation β on αO(G, τ ) is said to be α-open [9] if for every α-open set U of x ∈ G, there, exists an α β -open set V of G such that x ∈ V and V ⊆ U β .…”

“…The operation id : αO(G, τ ) → P (G) is defined by id(V ) = V for any set V ∈ αO(G, τ ) this operation is called the identity operation on αO(G, τ ) [9]. An operation β : αO(G) → P (G) is said to be α-monotone on αO(G) [2] if for all A, B ∈ αO(G), A ⊆ B implies A β ⊆ B β . An operation β : αO(G) → P (G) is said to be α-idempotent on αO(G) [2] if A ββ = A β for all A ∈ αO(G).…”

“…An operation β : αO(G) → P (G) is said to be α-monotone on αO(G) [2] if for all A, B ∈ αO(G), A ⊆ B implies A β ⊆ B β . An operation β : αO(G) → P (G) is said to be α-idempotent on αO(G) [2] if A ββ = A β for all A ∈ αO(G).…”

WEAKLY $\alpha_{\gamma}$-REGULAR AND WEAKLY $\alpha_{\gamma}$-NORML SPACES

“…The αCl γ -closure of A is denoted by αCl γ (A). The union of all α γ -open sets contained in A is called the α γ -interior of A and denoted by α γ Int(A) [3]. An operation γ on αO(X, τ ) is said to be α-regular [5] if for every α-open sets U and V of each x ∈ X, there exists an α-open set W of x such that W γ ⊆ U γ ∩ V γ .…”

α-ℽ-convergence, α-ℽ- accumulation and α-ℽ- compactness

“…An operation : O(X) ! P (X) is said to be -monotone [3] if for all A; B 2 O(X), A B implies A B . The operation id : O(X; ) !…”